server goes down frequently

I have a 64 bit Centos 5.5 with Apache 2.0.63 server, two postfix instances and Phusion Passenger.

The issue I have is that the server goes down frequently like about 10 times a day and takes about 10 minutes whenever it goes down before it wakes up on its own.

Now the strange thing is if I am logged to the server via SSH the server does not go down and the websites it serves continue running normally.
If the server is down and I access the server's WHM site via the browser, that is port 2087, the server wakes up immediately.

Does someone have an idea on what is going on and what I need to do to fix the issue?

I have checked all the logs that I think will be helpful but I am not getting any clue.
that is Apache error_log, access_log, maillog, mailq, messages, and secure

On Postfix,
I have 2 websites on the server and each with its own IP address. I wanted to set exim email server so that the emails sent can appear as if coming from two mail servers since a lot of email providers will reject emails if the domain name in the "from" field does not correspond to the domain's IP address. But I did not see how this could be done in exim so I ended up installing postfix since I was more familiar with it. I would be glad to use exim if it can be able to do this so that cPanel can manage it.
I will do what you have suggested.